Excursion operator - Knowlpedia

Let $F$ be a global function field and $G$ a connected reductive group. An excursion operator is indexed by

\left(I,f,(\gamma_i)_{i\in I}\right),

where $I$ is a finite set, $f\in\mathcal O(\widehat G\backslash\widehat G^I/\widehat G)$ is an invariant function on powers of the dual group, invariant under diagonal left and right translation, and $\gamma_i$ lies in the absolute Galois group $\operatorname{Gal}(\overline F/F)$ . It defines an endomorphism

S_{I,f,(\gamma_i)}

of the finite-level space of cuspidal automorphic forms.

Creation, excursion, annihilation

Choose a finite-dimensional representation $W$ of $\widehat G^I$ , a diagonal- $\widehat G$ -invariant vector $x\in W$ , and an invariant covector $\xi\in W^*$ such that

f((g_i))=\langle\xi,(g_i)x\rangle.

The operator is the composite:

create $I$ coincident shtuka legs using $x:\mathbf 1\to W$ ;
act on the legs by the independent Galois elements $(\gamma_i)$ , using partial Frobenius and Drinfeld's lemma;
annihilate the legs using $\xi:W\to\mathbf 1$ .

Coalescence of legs supplies the comparison maps. The resulting operator depends only on $I,f,(\gamma_i)$ , not on the chosen realization $(W,x,\xi)$ .

Why many legs are necessary

Characters of single dual-group elements do not distinguish all semisimple homomorphisms into a general reductive group. Simultaneous invariant functions on $\widehat G^I$ for all finite $I$ retain the pseudocharacter data needed to reconstruct a semisimple $\widehat G$ -valued Galois parameter.

Relations

Excursion operators are continuous in the Galois variables, compatible with maps of finite sets, multiplicative in invariant functions, and commute with one another and with unramified Hecke operators. They generate the excursion algebra.

References

Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” §§0.1 and 5–10. arXiv.
Vincent Lafforgue, “Shtukas for reductive groups and Langlands correspondence for function fields,” 2018. arXiv.